1. Digital Logic

2. 2 Abstractions in CS (gates) Basic Gate: Inverter IO 0101 1010 IO GNDI O Vcc Resister (limits conductivity) Truth Table

3. 3 Abstractions in CS (gates) IO 0101 1010 IO Truth Table

4. 4 Abstractions in CS (gates) Basic Gate: NAND (Negated AND) AB 00110011 01010101 11101110 Y ABAB Y GNDABY Vcc Truth Table

5. 5 Abstractions in CS (gates) Basic Gate: AND AB 00110011 01010101 00010001 Y ABAB Y Truth Table

6. 6 Abstractions in CS (gates) Other Basic Gates: OR gate AB 00110011 01010101 01110111 Y ABAB Y Truth Table

7. 7 Abstractions in CS (gates) Other Basic Gates: XOR gate AB 00110011 01010101 01100110 Y ABAB Y Truth Table

8. 8 Logic Blocks Logic blocks are built from gates that implement basic logic functions – Any logical function can be constructed using AND gates, OR gates, and inversion.

9. Adder In computers, the most common task is add. In MIPS, we write “ add $t0, $t1, $t2.” The hardware will get the values of $t1 and $t2, feed them to an adder, and store the result back to $t0. So how the adder is implemented?

10. Half-adder How to implement a half-adder with logic gates? A half adder takes two inputs, a and b, and generates two outputs, sum and carry. The inputs and outputs are all one-bit values. a bcarry sum

11. Half-adder First, how many possible combinations of inputs?

12. Half-adder Four combinations. absumcarry 00 01 10 11

13. Half-adder The value of sum should be? Carry? absumcarry 00 01 10 11

14. Half-adder Okay. We have two outputs. But let’s implement them one by one. First, how to get sum? Hint: look at the truth table. absum 000 011 101 110

15. Half-adder Sum a b sum

16. How about carry? The truth table is abcarry 000 010 100 111

17. Carry So, the circuit for carry is a b carry

18. Half-adder Put them together, we get a b sum carry

19. 19 1-Bit Adder 1-bit full adder – Also called a (3, 2) adder

20. 20 Constructing Truth Table for 1-Bit Adder

21. 21 Truth Table for a 1-Bit Adder

22. Sum? Sum is `1’ when one of the following four cases is true: – a=1, b=0, c=0 – a=0, b=1, c=0 – a=0, b=0, c=1 – a=1, b=1, c=1

23. Sum The idea is that we will build a circuit made of and gates and or gate faithfully according to the truth table. – Each and gate corresponds to one ``true’’ row in the truth table. The and gate should output a ```1’’ if and only if the input combination is the same as this row. If all other cases, the output is ``0.’’ – So, whenever the input combination falls in one of the ``true’’ rows, one of the and gates is ``1’’, so the output of the or gate is 1. – If the input combination does not fall into any of the ``true’’ rows, none of the and gates will output a ``1’’, so the output of the or gate is 0.

24. 24 Boolean Algebra We express logic functions using logic equations using Boolean algebra – The OR operator is written as +, as in A + B. – The AND operator is written as ·, as A · B. – The unary operator NOT is written as. or A’. Remember: This is not the binary field. Here 0+0=0, 0+1=1+0=1, 1+1=1.

25. Sum

26. Carryout bit? Carryout bit is ‘1’ also on four cases. When a, b and carryin are 110, 101, 011, 111. Does it mean that we need a similar circuit as sum?

27. Carryout bit Actually, it can be simpler

28. Delay Hardware has delays. Delay is defined as the time since the input is stable to the time when the output is stable. How much delay do you think the one-bit half adder is and the one-bit full adder is?

29. 29 1-Bit Adder

30. 32-bit adder How to get the 32-bit adder used in MIPS?

31. 32-bit adder